Page 301 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

9.6 Uniform Asymptotic Approximations 287

It follows that
∞ ∞

1 x z 2
exp{−z(y − log(y))} dy = exp(−z) exp − x dx
y x(c) y(x) − 1 2
c
√
(2π) ∞ x √ √
= √ exp(−z) zφ( zx) dx
z x(c) y(x) − 1
where
1
x(c) = sgn(c − 1){2[c − log(c) − 1]} 2
and y(x) solves
1 2
y − log(y) = x + 1.
2
By Theorem 9.16,
x √ √ √ 1
∞

zφ( zx) dx = [1 − ( zx(c))] h(0) + O
x(c) y(x) − 1 z
x(c)/(c − 1) − h(0) √
+ √ φ( zx(c))
zx(c)
where
1
x [2(y − log(y) − 1)] 2
h(0) = lim = lim = 1.
x→0 y(x) − 1 y→1 |y − 1|
Hence,
∞

x √ √ √ 1
zφ( zx) dx = [1 − ( zx(c))] 1 + O
x(c) y(x) − 1 z
x(c)/(c − 1) − 1 √
+ √ φ( zx(c))
zx(c)
and
√
1 (2π) z √ 1
Pr(X ≥ cE(X)) = √ exp(−z)z [1 − ( zx(c))] 1 + O
(z) z z
x(c)/(c − 1) − 1 √
+ √ φ( zx(c)).
zx(c)
Using Stirling’s approximation for (z)in this expression yields

√ 1
Pr(X ≥ cE(X)) = [1 − ( zx(c))] 1 + O
z
x(c)/(c − 1) − 1 √
+ √ φ( zx(c)) as z →∞.
zx(c)
For comparison, we may consider approximations based on Laplace’s method. The
function g(y) = log(y) − y is maximized at y = 1 and is strictly decreasing on the interval
(1, ∞). Hence, if c < 1, we may use the approximation given in Theorem 9.14, leading to
the approximation
z−1
z 2 √ −1
Pr(X ≥ cE(X)) = exp(−z) (2π)[1 + O(z )]. (9.6)
(z)